Theorem anifp 1020

Description: The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 1021. (Contributed by BJ, 30-Sep-2019.)

Assertion

Ref	Expression
anifp	⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))

Proof of Theorem anifp

Step	Hyp	Ref	Expression
1		olc 399	. . 3 ⊢ (𝜓 → (¬ 𝜑 ∨ 𝜓))
2		olc 399	. . 3 ⊢ (𝜒 → (𝜑 ∨ 𝜒))
3	1, 2	anim12i 590	. 2 ⊢ ((𝜓 ∧ 𝜒) → ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
4		dfifp4 1016	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))
5	3, 4	sylibr 224	1 ⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384  if-wif 1012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013

This theorem is referenced by:  bj-consensus  32562  bj-consensusALT  32563  axfrege58a  38168